Robust Difference-in-differences Models

TL;DR

This paper introduces a robust generalized difference-in-differences method that leverages multiple data sources and pre-treatment information to improve causal inference, relaxing the traditional parallel trends assumption.

Contribution

It develops a new approach interpreting parallel trends via selection bias, allowing for multiple pre-treatment periods and covariates, and provides an identified set for the treatment effect.

Findings

The method produces an identified set containing the true ATT.

It extends to multiple treatment periods settings.

Numerical and empirical examples demonstrate its effectiveness.

Abstract

The difference-in-differences (DID) method identifies the average treatment effects on the treated (ATT) under mainly the so-called parallel trends (PT) assumption. The most common and widely used approach to justify the PT assumption is the pre-treatment period examination. If a null hypothesis of the same trend in the outcome means for both treatment and control groups in the pre-treatment periods is rejected, researchers believe less in PT and the DID results. This paper develops a robust generalized DID method that utilizes all the information available not only from the pre-treatment periods but also from multiple data sources. Our approach interprets PT in a different way using a notion of selection bias, which enables us to generalize the standard DID estimand by defining an information set that may contain multiple pre-treatment periods or other baseline covariates. Our main assumption states that the selection bias in the post-treatment period lies within the convex hull of all selection biases in the pre-treatment periods. We provide a sufficient condition for this assumption to hold. Based on the baseline information set we construct, we provide an identified set for the ATT that always contains the true ATT under our identifying assumption, and also the standard DID estimand. We extend our proposed approach to multiple treatment periods DID settings. We propose a flexible and easy way to implement the method. Finally, we illustrate our methodology through some numerical and empirical examples.